A generalization of Hasse's generalization of the Syracuse algorithm

Matthews, Keith R.; Watts, A. M.

doi:10.4064/aa-43-2-167-175

Cited by 20 publications

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“…Therefore g is cyclical on S with period 7' = V = 5, and every B(M, z) E O(gls) is a cyclical class. Moreover {3 = t(1, 1, 1, 1, 1), and by (33) Pr = t for r E {a, 1,2,3,6} = SR (9). Therefore by (10), a(gls) = 1 and Theorem 3 holds.…”

Section: Explanatory Examplesmentioning

confidence: 84%

“…A survey of literature and results concerning the 3x + 1 problem and its generalizations has been given by Lagarias in [7]. In particular, in [9,10] Matthews and Watts have studied the functions g E G], and in [8] Leigh has studied all functions g E G. They have obtained some partial results, which have been improved in [11] for the functions g E Go only.…”

Section: Introductionmentioning

confidence: 99%

“…The fixed vector is f3= 1~I(54,3,29,9,3,3). By (33) Po= tf3o= 1~1' PI = tf3o+ if31 + if33= 1~1' P2 = tf30 + !f32 + f34 = igz, and so on.…”

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confidence: 99%

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Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

Venturini

1992

Stud Appl Math

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Given pEN = {1, 2, 3 ... }, as a generalization of the 3x + 1 problem [7], we study the behavior of the sequences s(m) = {mn}n ~ 0' mE Z (the set of the integers), defined by the iterative formulawhere r == m n (mod p) , r E R(p) ={O,I, ... ,p-l}, and a r = t r / p (r r EN), and b r are chosen in such a way that m n E Z for every n. Our aim is to establish when these sequences are divergent, or convergent into a cycle, considering also a fixed point as a cycle. Moreover, the structure of possible cycles is investigated.Definition 1'), it follows that ts. = d s = p for every j. Finally, since S is a g-ergodic set, there exists a cycli~al p~rmutation 1T of S (') R( p) such that g I s can be written in the formwhich is (11) for k = 1. • Remark: If in (11) k > 1, then 1T may not be a cyclical permutation, since the g-ergodic set S may not be a gk-ergodic set (again see Examples 4 and 5). 2.4When a = a(gls) < 1, as in [11, Lemma 1], we have LEMMA 2. For every Sj E SR(p), ifin (1), b s = 0, then put as = as. and k j = 0, while if b Sj -=1= 0, then put a Sj = a Sj + 1/ pk j , k; EN. Then for every b Sj -=1= ° there exists k· E N such that a < a = a Po ••• a Ph -1 < 1. J So 'h-l

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Section: Explanatory Examplesmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

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Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

Venturini

1992

Stud Appl Math

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“…Note that Conjecture 1.2(i) does not hold for some generalizations of the 3x+1 problem studied by Möller, Matthews and Watts in [4,5] ; Conjecture 1.2(ii) is a special case of Conjecture (iv) in [4]; Conjecture 1.2(iii) implies that there is some n such that 2 bn > 3 n in the E-sequence (a n ) n⩾1 of every odd positive integer x, which is a conjecture posed by Terras in [8] about his τ −stopping time.…”

Section: Introductionmentioning

confidence: 99%

An E-Squence Approach to the 3x + 1 Problem

Wang¹

2019

Preprint

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For any odd positive integer x, define (x n ) n⩾0 and (a n ) n⩾1 of all 2 an is called the E-sequence of x. For example, the orbit and the E-sequence of 3 are (3, 5, 1, 1, . . .) and (1, 4, 2, 2, . . .), respectively.Given any sequence (a n ) n⩾1 of positive integers, if it is the E-sequence of the odd positive integer x, it is called to be convergent to x and, denoted by Ω − lim a n = x; if (a n ) n⩾1 is not the E-sequence of any odd positive integer, 2010 Mathematics Subject Classification. Primary 11A99; Secondary 11B83.

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“…What we are concerned with is the Ω-convergence and Ω-divergence of any infinite sequence of positive integers, i.e., the generalized E-sequences. Note that Conjecture 2(i) does not hold for some generalizations of the 3x + 1 problem studied by Möller, Matthews, and Watts in [12,13]; Conjecture 2(ii) implies that there is some n such that 2 b n > 3 n in the E-sequence (a n ) n⩾1 of every odd positive integer x, which is a conjecture posed by Terras in [11] about his τ-stopping time.…”

Section: Introductionmentioning

confidence: 99%

An E-Sequence Approach to the 3x + 1 Problem

Wang

2019

Symmetry

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For any odd positive integer x, define ( x n ) n ⩾ 0 and ( a n ) n ⩾ 1 by setting x 0 = x , x n = 3 x n - 1 + 1 2 a n such that all x n are odd. The 3 x + 1 problem asserts that there is an x n = 1 for all x. Usually, ( x n ) n ⩾ 0 is called the trajectory of x. In this paper, we concentrate on ( a n ) n ⩾ 1 and call it the E-sequence of x. The idea is that we generalize E-sequences to all infinite sequences ( a n ) n ⩾ 1 of positive integers and consider all these generalized E-sequences. We then define ( a n ) n ⩾ 1 to be Ω -convergent to x if it is the E-sequence of x and to be Ω -divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the Ω -divergence of all non-periodic E-sequences implies the periodicity of ( x n ) n ⩾ 0 for all x 0 . The principal results of this paper are to prove the Ω -divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences ( a n ) n ⩾ 1 with lim ¯ n → ∞ b n n > log 2 3 are Ω -divergent by using Wendel’s inequality and the Matthews and Watts’ formula x n = 3 n x 0 2 b n ∏ k = 0 n - 1 ( 1 + 1 3 x k ) , where b n = ∑ k = 1 n a k . These results present a possible way to prove the periodicity of trajectories of all positive integers in the 3 x + 1 problem, and we call it the E-sequence approach.

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A generalization of Hasse's generalization of the Syracuse algorithm

Cited by 20 publications

References 0 publications

Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

Iterates of Number Theoretic Functions with Periodic Rational Coefficients (Generalization of the 3x + 1 Problem)

An E-Squence Approach to the 3x + 1 Problem

An E-Sequence Approach to the 3x + 1 Problem

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